Problem 22 - Precalculus: Functions and Graphs

Question

Verify the Identity by expanding each determinant. $$\left|\begin{array}{ll} a & b \\ c & d \end{array}\right|=-\left|\begin{array}{ll} b & a \\ d & c \end{array}\right|$$

Step-by-Step Solution

Answer

The identity is verified as both sides simplify to \( ad - bc \).

Step 1: Understand the Determinant of a 2x2 Matrix

For any 2x2 matrix \( \begin{bmatrix} p & q \ r & s \end{bmatrix} \), the determinant is calculated as \( ps - qr \). This formula will help us expand the determinants in the given identity.

Step 2: Expand the Left-hand Determinant

Expand the determinant \( \left| \begin{array}{cc} a & b \ c & d \end{array} \right| \). The formula provides: \( ad - bc \).

Step 3: Expand the Right-hand Determinant

The determinant \( \left| \begin{array}{cc} b & a \ d & c \end{array} \right| \) expands to \( bc - ad \) using the same determinant formula.

Step 4: Compare Both Sides of the Identity

The identity states that the left determinant's expansion \( ad - bc \) is equal to the negative of the right determinant's expansion, i.e., \(-1 \times (bc - ad)\). Distribute the negative sign: \(-bc + ad\).

Step 5: Verify the Identity

Compare the expressions derived in step 2 and step 4: \( ad - bc = ad - bc \). Both sides are equal, verifying the identity is correct.

Key Concepts

2x2 Matrix

A 2x2 matrix is one of the simplest forms of a matrix. It refers to a matrix that has two rows and two columns. You can think of it as a grid with four elements. For example, consider \[ \begin{bmatrix} a & b \ c & d \end{bmatrix} \].This represents a general form of a 2x2 matrix with elements labeled as follows:

• "a" in the first row and first column,
• "b" in the first row and second column,
• "c" in the second row and first column,
• "d" in the second row and second column.

Matrices are very useful in mathematics and various fields like physics, computer science, and engineering. They help solve systems of equations, perform transformations, and much more.

Determinant Expansion

Expanding or calculating the determinant of a 2x2 matrix is straightforward and very important. The formula to find the determinant of a 2x2 matrix \[ \begin{bmatrix} p & q \ r & s \end{bmatrix} \] is given by:\[ \text{determinant} = ps - qr \]This formula means you multiply the top left element, "p," by the bottom right element, "s," and subtract from this the product of the elements "q" and "r" from the top right and bottom left, respectively.

This calculation is crucial in various mathematical operations because it gives a scalar value that provides important information about the matrix, such as invertibility. A nonzero determinant indicates that the matrix is invertible, meaning it has an inverse.

Matrix Identity Verification

Matrix identity verification involves proving that two matrix expressions are indeed equal. It's an important concept used to verify mathematical claims or properties involving matrices. In the exercise, we are comparing two determinants to confirm they are negatives of each other.The given identity:\[ \left|\begin{array}{ll} a & b \ c & d \end{array}\right| = -\left|\begin{array}{ll} b & a \ d & c \end{array}\right| \]implies that the number from the determinant on the left, calculated as \( ad - bc \), must equal the negative of the determinant on the right, \(-(bc - ad)\), which simplifies to \( ad - bc \).

This verification essentially proves that switching rows and columns in a 2x2 matrix flips the sign of its determinant. By expanding and comparing both determinants, you can see both sides of the identity are equivalent, confirming the relation holds true.